Jana Pócsová
3. On relationship between incidence relations and generalized one-sided concept lattices
Figure 1: Generalized one-sided concept lattice Proposition 2.5. The set C B, A,L, R
with the partial order defined by (2.7) forms a complete lattice, where
^
Proof of this proposition is based on the fact that any Galois connection be-tween complete lattices induces dually isomorphic closure systems (see [13]). Con-sequently, this dual isomorphism maps infima on the one side onto suprema in a closure system on the other side and vice versa.
Remark that the algorithm for generation of generalized one-sided concept lat-tices can be found in [7] or [8].
The Hasse diagram of the generalized one-sided concept lattice determined by Table 1 is shown on Figure 1. Let us remark that we denote the elements of direct product as ordered tuples, as it is common in lattice theory.
3. On relationship between incidence relations and generalized one-sided concept lattices
In this section we present our results concerning incidence relations and correspond-ing one-sided concept lattices. We also describe the order structure of the set of all mappings involving in some Galois connection between power set and the direct product of complete lattices. Firstly, we show that the correspondence
generalized one-sided context 7→ generalized one-sided concept lattice
is injective or equivalently one-to-one. We already know how to define generalized one-sided lattice from given formal context. However, there is an interesting theo-retical question, whether different formal contexts yield different one-sided concept lattices. The positive answer means that not only formal context fully characterizes generalized one-sided context, but the converse is also true, i.e., given generalized sided concept lattice fully determines formal context. Hence, generalized one-sided concept lattice contains all information about object-attribute model.
We recall the definition of injective mapping. A mappingf :A→B is said to be injective (one-to-one) if
x6=y implies f(x)6=f(y)
Evidently, this condition is equivalent to the condition f(x) =f(y)impliesx=y.
In what follows, we will consider that the set of objectsBis fixed, as well as the set of all attributesA (together with truth value structuresL(a)). Consider that we have two generalized one-sided formal contexts(B, A,L, R1)and(B, A,L, R2).
The corresponding concept lattices are denoted byC1=C B, A,L, R1
andC2= C B, A,L, R2
.
Theorem 3.1. The correspondence(B, A,L, R)7→ C B, A,L, R, which assign to each generalized one-sided formal context the corresponding generalized one-sided concept lattice is injective.
Proof. We prove this theorem in two steps. Firstly we show that the correspondence (B, A,L, R)7→(↑,↓), which maps formal context onto the Galois connection given by (2.5) and (2.6) respectively, is injective. Next we show that the correspondence (↑,↓) 7→ C B, A,L, R
, which maps Galois connection to the concept lattice is injective too. Since the composition of two injective mappings is injective, this will satisfy to prove our result.
Suppose that incidence R1 and R2 differ, i.e., there exist b ∈ B, a ∈ A such that R1(b, a) 6= R2(b, a). Note, that we will recognize the corresponding Galois connection by subscript. According to the definition (2.5) of mapping↑we obtain:
↑1({b}) = ^
b0∈{b}
R1(b0, a) =R1(b, a)6=R2(b, a) = ^
b0∈{b}
R2(b0, a) =↑2({b}).
This equation shows that we have found one-element subset {b} with ↑1({b}) 6=
↑2({b})and consequently (↑1,↓1)6= (↑2,↓2). Hence, the first correspondence be-tween formal contexts and Galois connections is injective.
Further, assume that C1 = C2, i.e., that the generalized one-sided concept lattices equal. This means that the sets of fixed points coincide, i.e., for allX ⊆B andg∈Q
a∈AL(a)it holds
↑1(X) =gand↓1(g) =X iff ↑2(X) =gand↓2(g) =X. (3.1) LetX ⊆B be an arbitrary subset. From the property (2.2) of Galois connec-tions we have ↑1(X) =↑1(↓1(↑1(X))), thus ordered pair(↓1(↑1(X)),↑1(X))forms
a fixed point of Galois connection(↑1,↓1). Then, due to condition (3.1) we obtain that ↓2(↑1(X)) =↓1(↑1(X)). Consequently, we haveX ⊆ ↓1(↑1(X)) =↓2(↑1(X)) which yields the first half of the condition (c) of the Definition 2.2.
Similarly, using (2.2) we obtain for each element g ∈ Q
a∈AL(A) the pair (↓2(g),↑2(↓2(g)) forms fixed point of (↑2,↓2). Again, due to condition (3.1) we obtain ↑2(↓2(g)) = ↑1(↓2(g)), which yields g ≤ ↑2(↓2(g)) = ↑1(↓2(g)). Since the mappings ↑1 and ↓2 are order reversing, we have proved that the pair (↑1,↓2) forms Galois connection. Now using the fact that dual adjoint is unique, we obtain
↑1=↑2 and↓1=↓2, which completes the proof.
It was proved in [7] that for any Galois connection(Φ,Ψ) betweenP(B) and Q
a∈AL(a) there exists a generalized formal context (B, A,L, R)that ↑ = Φ and
↓ = Ψ. Hence the correspondence between formal contexts and generalized one-sided concept lattices is surjective, too. Since we have shown that it is injective, in fact this correspondence is bijective. Using this fact we can prove the following theorem about number of all concept lattices.
Theorem 3.2. Let B 6= ∅ be set of objects, A = {a1, a2, . . . , am} be set of at-tributes. Denote by n=|B| number of objects and for all i= 1, . . . , m denote by ni=|L(ai)|the cardinality of the complete lattice L(ai). Then there is(Qm
i=1ni)n generalized one-sided concept lattices.
Proof. There is a bijection between set of all generalized incidence relations and one-sided concept lattices, thus it is sufficient to count all generalized incidence relations. For each object b and each attribute a the value R(b, a) can obtain ni =|L(ai)|values. Since we haven objects, there isnin possibilities for columns in data table (which represents incidence relation). Together we have
n1n possibilities to define incidence relation.
This result generalizes the similar assertion for classical concept lattices. Sup-pose there is given a formal context(B, A, I). If we havenobjects andmattributes, then there is2n·mconcept lattices. Any classical concept lattice can be character-ized as generalcharacter-ized one-sided concept lattice by settingL(a) =2(2={0,1}denotes two-element chain) and R(b, a) = 1 if and only if(b, a)∈ I (see [14] for details).
Hence applying the result of Theorem 3.2 we obtainQm
i=12n = (2n)m= 2m·n. Similarly, if one will considerL(ai) =L for all i= 1, . . . , m, than generalized one-sided concept lattices, represent one-sided concept lattices. Hence, applying Theorem 3.2 we obtain that there isQm
i=1|L|n=|L|m·ndifferent one-sided concept lattices.
Next we show that formal contexts also characterize order properties of the Galois connections between power sets and complete lattices. Firstly we prove the following lemma, concerning the closure property of Galois connections. LetLand
M be complete lattices. Denote byGal(L, M)the set of all ϕ:L →M such that there existsψ:M→Ldually adjoint to ϕ.
Lemma 3.3. Let L, M be complete lattices. The set Gal(L, M) forms a closure system in complete latticeML.
Proof. We show that the setGal(L, M)is closed under arbitrary infima. Let{ϕi: i∈I} ⊆Gal(L, M) be an arbitrary system. Denote byϕ=V
SinceGal(L, M) forms a closure system in complete latticeML, it forms com-plete lattice too. In this case meets in Gal(L, M)coincide with the meets inML, but this is not valid for joins in general. In particular, if(ϕi:i∈I} ⊆Gal(L, M) where the symbolsV andWdenote operations of meet and join inML.
Let us note thatGal(L, M)andGal(M, L)forms isomorphic posets. This follows from the fact that the correspondence ϕ 7→ ψ where ψ denotes the dual adjoint of ϕ is bijective. Moreover it is order preserving in both directions. Suppose ϕ1(x) ≤ ϕ2(x) for all x ∈ L. Let y ∈ M be an arbitrary element. Then y ≤ ϕ1(ψ1(y)) ≤ ϕ2(ψ1(y)) and according to the condition (2.1) it follows ψ1(y) ≤ ψ2(y). The opposite implication can be proved analogously, hence ϕ1≤ϕ2 if and only ifψ1≤ψ2.
Further assume that B, A6=∅ and L:A→CLare fixed. In order to describe the structure of the lattice Gal(P(B),Q
a∈AL(a)) we denote by R(B, A,L) the set of all relations R such that (B, A,L, R) forms generalized one-sided formal
context. Obviously the set R(B, A,L) forms complete lattice. In this case, if {Ri:i∈I} is a system of relations, then relationR whereR(b, a) =V
i∈IRi(b, a) (R(b, a) =W
i∈IRi(b, a)) corresponds to the infimum (supremum).
Theorem 3.4. The lattice Gal(P(B),Q
a∈AL(a)) is isomorphic to the lattice of all incidence relationsR(B, A,L).
Proof. Define F: R(B, A,L) → Gal(P(B),Q
a∈AL(a)) for all R ∈ R(B, A,L) by F(R) =↑R, where ↑R is defined by (2.5). As we already know, the mapping F is bijective. We show, that it also preserves the lattice operations, i.e.,F R1∧R2
=
Hence the mappingF preserves meets.
In order to prove thatF preserves joins, we use the fact that the mapping F is surjective, i.e., for any Galois connection (ϕ, ψ)between P(B) and Q
a∈AL(a)
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